Let \(K\)  be the positive cone in \(E\)  consisting of all non-negative continuous functions. A natural length function \(\ell \)  for \(\varrho ^{ }_{a}\)  is a function in \(K\)  for which \(M\ell =\lambda \ell \)  for some \(\lambda \ne 0\) . In other words, it corresponds to a non-negative eigenvector of \(M\)  in \(K\)  with non-zero eigenvalue \(\lambda \) .
If further \(\lambda >1\)  and \(\ell \)  is strictly positive, i.e., \(\ell (b)>0\)  for all \(b\in \mathcal {A}\) , one can associate a geometric inflation rule to \(\varrho ^{ }_{a}\)  which generates substitution tilings on \(\mathbb {R}\)  with (possibly) infinitely many prototile lengths. The following result is proved in [1]}.
